Parameter Estimation from Single Patient, Single Time-Point Sequencing Data of Recurrent Tumors

Abstract

In this study, we develop consistent estimators for key parameters that govern the dynamics of tumor cell populations when subjected to pharmacological treatments. While these treatments often lead to an initial reduction in the abundance of drug-sensitive cells, a population of drug-resistant cells frequently emerges over time, resulting in cancer recurrence. Samples from recurrent tumors present as an invaluable data source that can offer crucial insights into the ability of cancer cells to adapt and withstand treatment interventions. To effectively utilize the data obtained from recurrent tumors, we derive several large number limit theorems, specifically focusing on the metrics that quantify the clonal diversity of cancer cell populations at the time of cancer recurrence. These theorems then serve as the foundation for constructing our estimators. A distinguishing feature of our approach is that our estimators only require a single time-point sequencing data from a single tumor, thereby enhancing the practicality of our approach and enabling the understanding of cancer recurrence at the individual level.

Figures (5)

• Figure 1: In this experiment, we set $\alpha = 0.5, \lambda_0 = -0.5, r_1 = 1.5, d_1 = 1$ and $\lambda_1 = 0.5$. We let $n$ vary from $100$ to $100\times 2^{20}$. For each value of $n$, we repeat the experiment $k = 20$ times. We represent the mean relative errors of each estimator with a dark blue line and illustrate the standard deviation using a shaded area in light blue.
• Figure 2: In this experiment, we conduct a comparative analysis across three experimental setups. We set $\alpha = 0.8$, $\lambda_0 = -0.2$, $\lambda_1 = 0.8$ and $r_1 = 2.0$. The first setup is a baseline scheme without restrictions on clone size observation; the second setup imposes a threshold, only considering clones that constitute at least $2\%$ of the total population; and the third raises this threshold to $10\%$. We assess the impact of these varying observation restrictions on the performance of the estimators for $\lambda_1$ and $\alpha$.
• Figure 3: In this experiment, we vary the model parameters and check the stability of our estimators. In the first setup, We fix $r_1 = 1.5$, $d_1 = 1.0$, and $\alpha = 0.5$, and select $\lambda_0$ uniformly from the interval $(-0.9, -0.1)$ (see the first row of Figure \ref{['fig:random_para']}). In the second setup, we fix $r_0 = 1.0$, $d_0 = 1.5$, and $\alpha = 0.5$ and select $\lambda_1$ uniformly from the interval $(0.3, 0.9)$ (see the second row of Figure \ref{['fig:random_para']}). Lastly, we set $r_0 = d_0 = 1.0$ and $r_1 = d_1 = 1.5$, and select $\alpha$ uniformly from the interval $(0, 1)$ (see the third row of Figure \ref{['fig:random_para']}). Under all setups, we fix the initial tumor burden to be $n = 1 \times 10^6$, and conduct numerical simulations to estimate $\lambda_0$, $\lambda_1$, $\alpha$, and $r_1$. The resulting relative errors for these estimations, corresponding to each of the defined setups, are presented in Figure \ref{['fig:random_para']}.
• Figure 4: In this experiment, we set $n=10^7$, $\alpha = 0.8$, $\lambda_0 = -0.2$, and $\lambda_1 = 0.8$, and conduct $10$ experiments. We first derive a set of estimators using the approach developed in this paper, after which we employed bootstrapping techniques to refine these estimates. To facilitate a visual comparison of the bootstrapping technique's effectiveness, we generated violin plots illustrating the distribution of the ratio of relative errors post-bootstrapping to those pre-bootstrapping. The median of these distributions is highlighted by a red line.
• Figure 5: In this experiment, we set $n=10^5$, $\alpha = 0.5$, $\lambda_0 = -0.5$, and $\lambda_1 = 0.5$. We vary the capacity $C$ from $2n$ to $5n$ and record the relative error of the estimators across different capacity settings.

Theorems & Definitions (39)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Theorem 1: Convergence rate of $\gamma_n$
• proof
• Theorem 2: Convergence of $I_n(\gamma_n)$
• proof
• Proposition 1